Surface
Contents
Author Index
Qingyuan LI, Yanqing CHANG & Daiyong CAO
BOUNDARY, COBOUNDARY, SIX GROUPS TOPOLOGY AND ‘ONE FACE WITH
THREE LAYERS' IN 3D GIS
Qingyuan LI a,b, Yanqing CHANGa, Daiyong CAOb
a
Chinese Academy of Surveying&Mapping, Beijing Beitaiping Road 16, 100039
LiQY@casm.ac.cn
b
The Key Laboratory of Coal Resource, Ministry of Education, China, Institute Road Ding 11,Beijing ,100083
Commission II, WG II/3
KEY WORDS:
3DGIS, Boundary, Coboundary, Topology, One face with three layers
ABSTRACT:
On recognition to boundary and coboundary in ISO TC211 Geographic Information “Spatial schema” International Standard. The
author modify ‘five-group relations’, which the authors have proposed formerly, to ‘six-group relations’. A new concept of 'One
face with three layers' in 3D GIS topological relation is given in the paper. It is a supplement and development for relation of
‘Face-ExteriorRing-InteriorRing-PositiveSolid-NegativeSolid’ in the ‘Five group relations’. 'One face with three layers' supposes
that one face has three layers, i.e. positive layer, middle layer and negative layer. Positive layer and negative layer are divided
separately, middle layer overlap division of positive layer and negative layer. In this way, the integrality of a nature face is retained,
division of positive side and negative side are isolated, but have some connection through middle layer. Data structure of ‘One face
with three layers’ is designed in the paper. If construction 3D topologic relation in this way, inquire of one polyhedron’s adjacent
polyhedrons is very easy. Last, the author expands the concept of “One face with three layers” to 2DGIS, to deal with the line object
like freeway, railway and river, which may be nature boundary of districts.
1. INTRODUCE
In spatial, the real world is three dimensions, but most current
GIS can not perfectly express multi-layer information in vertical,
so it is called 2D or 2.5D GIS. The domains of geology, mining,
environment and other applications need 3DGIS platform to
support processing real 3D information. Not all the applications
need 3DGIS. In fact, for most applications, 2D or 2.5D is
enough. The users who real need 3DGIS, may only 5% of GIS
users, but these applications have very important position in the
industry, it may produce 10% market gain. With the
development of 3D visualization tools, 3DGIS again become a
study hot point in GIS. In 3D GIS, there are many theory and
technique problems, which have not been solved perfectly. One
of them is 3DGIS’s data model and topological relationship.
Many scholars have studied this problem from different
standpoints. Martien Molenaar (1990) and Dieter Fritsch (1996)
studied formal data structure for 3D vector model. Han Guojian,
(1992), Guo Dazhi (Guo, et al., 1993) study linear octatree
express 3D underground object. Chen Jun & Guo Wei (Guo and
Chen, 1997, 1998) have studied in 3D geometry elements
topologic from 9 intersection model. Pilouk Morakot (1994),
Chen Xiaoyong (1994a, 1994b) study tetrahedron voxel fill
model. Li Qingyuan & Cao Daiyong(Li, 1996; Li, et al.,1996)
from mine & geology application, have studied in (1) ‘Five
group relation’ to express 3D object topologic relationship, (2)
‘Introducing surface - dividing body’ as a way to dynamic build
and maintain the 3D topology, (3) ‘One face with three layers’
to deal with contradiction of integrality of face with difference
of both side (positive, negative) adjoin polyhedron, (4) 3D
volume function interpolation in solid inner.
In ISO Geographic Information Spatial Schema (ISO/DIS
19107), a geographic spatial model is given. Boundary and
coboundary of spatial object is emphasized in topological
package. In this paper, on understanding to boundary and
265
coboundary, the authors review the five group relationships that
the author gave before. Lurked problem of ‘Face - ExteriorRing
- InteriorRings - PositiveSolid - NegativeSolid’ is discussed. A
new concept of 'One face with three layers' is given in the paper,
and extending it in 2D as ‘One line with three layers’.
2. BOUNDARY AND COBOUNDARY OF SPATIAL
ENTITY
Boundary and coboundary of geographic object are emphasized
in the ISO geographic information international standard
‘Spatial Schema’ (ISO/DIS 19107).
Boundary is a set that represents the limit of an object, i.e. a
transition between an object and the rest of its domain of
discourse. Dimension of an object boundary is less one than the
entity. For example, the boundary of a line (1-dimension) is a
set of two end points (0-dimension). Boundary of a face
(2-dimension) is a set of lines (1-dimension) which are grouped
to rings that one is exterior ring and some are interior rings (if
the face has holes). Boundary of a solid (3-dimension) is a set of
faces, which are grouped to one exterior shell and some are
interior shells (if the solid has holes). Coboundary is a set of
topological primitives of higher topological dimension
associated with a particular topological object, such that this
topological object is in each of their boundaries. If a geometry
object A is in the boundary of a geometry object B, then the B is
coboundary of A. For example, If a node is on the boundary of
an edge, that edge is on the coboundary of that node. Any
orientation parameter associated to one of these relations would
also be associated to the other. If an edge is on boundary of a
face, the face is coboundary of the edge. If a face is on
boundary of a solid, the solid is coboundary of the face.
IAPRS, VOLUME XXXIV, PART 2, COMMISSION II, Xi’an, Aug. 20-23, 2002
Spatial object can associate to the object which are less one
dimension and on its boundary by getting boundary operation,
also can associate to the object which is higher one dimension
than it and as the central object in a complex. In one dimension
linear case, coboundary of one node has at most two edges. In
the full topology case, there are precisely 2 edges. In the
2-dimensional planar case, the coboundary of an edge has at
most 2 faces. In the full topology case, there are precisely 2
faces, one face lying to the left of the edge, and another lying to
the right of the edge. In the 3-dimensional solid case, the
coboundary of face has at most 2 solids. In the full topology
case, there are precisely 2, one solid lying to the positive side of
the face, and another lying to the negative side of the face.
or negative agents of faces, which are grouped to one
ExteriorShell and some InteriorShell (if Solid have interior
hole). Coboundary of a Solid is empty.
These six group relations can be expressed by C language as
Appendex A.
Though in ISO/DIS 19107 said that a solid may has interior
boundary shells only, no exterior boundary shell, or a face has
interior boundary ring only, no exterior boundary ring. The
author thinks that in general GIS application, these special cases
could not be considered.
4. ‘ONE FACE WITH THREE LAYERS'
3. SIX GROUPS TOPOLOGICAL RELATION
4.1 Puzzle of Sub-Face Spliting
Although TC211 Spatial Schema has defined self-contained
vector spatial model, it is not oriented to program realization.
On the view of boundary and coboundary, the author think that
the five group of relation, which the author gave formerly, are
mainly considering of the spatial primitive and its boundary and
coboundary. If adding shell, the next six group relations could
be as a base of 3D GIS.
3D GIS is an expanding of 2D GIS in 3D space. Rechecking 2D
GIS’s topological relationship, which we are familiar with, has
help to we study 3D GIS, for we often can expand the think way
form 2D to 3D. Now, from a figure 1, let us check relation
‘Edge – BeginNode – EndNode – LeftPolygon – RightPolygon’
in 2D GIS.
(1) Node-BeginEdges-EndEdges: Node, an array of edges,
which begin from the node, and an array of edges, which end at
the node. The BeginEdges and EndEdges, all are coboundary of
the node.
(2) Edge-BeginNode-EndNode-Rings: Edge, begin and end
node of the edge, and array of rings which through the edge.
Two of the BeginNode and the EndNode are boundary of the
edge, interior adjacent faces of the rings are coboundary of the
edge.
Figure 1. In 2D GIS, an edge only has one left polygon and on
right polygon
(3) Ring-Edges-InteriorFace: Ring, an array of edges, which
composed the ring, and interior face of the ring. The edges are
composed component of the ring, not the boundary of the ring.
The ring is cycle, so the boundary is empty. The ring has
direction, interior adjoin face is on the left side of the ring. The
ring is a bridge between of face and the edges that are the
boundary composed component.
If the number of left or the right polygons of an edge are more
than one, the edge should be divided into several edges, to
ensure every edge only has one left polygon and one right
polygon. Recalling in 2D full topology ,coboundary of an edge
has just 2 polygons, one lying to the left and another lying to the
right of the edge. In figure1, the edge AD should be divided into
AB, BC, CD. In the same way, the relation of
‘Face-ExteriorRing-InteriorRings-PositiveSolid-NegativeSolid’
in 3D has the same case. If positive solid or negative solid of a
face are more than one, the face should be divided into several
sub-face, to make every sub-face only has one positive solid and
one negative solid. But the sub-dividing will bring other
problems.
(4)Face-ExteriorRing-InteriorRings-PositiveSolid-NegativeSoli
d: Face, ExteriorRing that composed exterior boundary,
InteriorRing that boundary interior holes (if they exist), positive
adjacent solid and negative adjacent solid. Because
ExteriorRing is most important and use frequently, so deal with
ExteriorRing and InteriorRing apart. It is also accepted that
combine the ExteriorRing and InteriorRing, and assume that
first ring ExteriorRing, afterward are InteriorRing. When the
number of ring is 1, without InteriorRing. Boundary of the face
is a set of edge that are on the rings (ExteriorRing +
InteriorRing). Coboundary of the face is a set of PositiveSolid
and NegativeSolid.
(5) Shell-Faces-InteriorSolid: Shell, an array of the faces that
composed the shell and the interior solid of the shell. A shell is
composed of many oriented face and negative references of
faces. A shell is a cycle, so its boundary is empty. A shell is a
bridge between Solid and the faces. A shell has direction,
interior solid is in interior of a shell. The direction of a shell
points to away from the solid.
Figure 2. The ground face should be divided into three
sub-faces
(6) Solid-ExteriorShell-InteriorShells: Solid, ExteriorShell and
an array of InteriorShell. The boundary of solid is a set of faces
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Qingyuan LI, Yanqing CHANG & Daiyong CAO
In figure 2, a simple geology model. Above the ground face,
east is water and west is land. Under the ground face, east is
modern deposit and west is base rock. The ground face seem to
be split into IQTLI、QMPTQ、MJKPM three sub-faces. Another
example is a fault face, in figure 3, seems to be divided into five
sub-faces.
Figure 5. Shore line divide positive layer into land area and
water area
The negative layer (IJKLI) is divided into two areas of base
rock area (IQTLI) and loose deposit area (QJKTQ) by the base
rock outcrop line QT (intersection line of unconformity face and
the ground face). It is show in figure 6.
Figure 3. The fault face should be divided into five sub-faces
An original nature face (ground face, fault face) has to be
divided into several sub-faces for building topologic relation. Is
it in reasonable? On the other hand, change of the adjacent solid
in positive side or negative side of a face will affect all division,
but actually, division in positive side and negative side has not
relation. We hope that integrality of a nature face should be
maintained, positive and negative side division should be
isolated each other, but should have some connection. So, it
leads to next concept of ‘One face with three layers’.
Figure 6. Base rock outcrop line divide negative line into base
rock area and loose deposit area
In the middle layer, the shore line MP and base rock outcrop
line QT overlap, divide the middle layer into three area of
IQTLQ, QMPTQ and MJKPM. Seeing Figure.7.
4.2 Concept of ‘One Face with Three Layers’
The concept of ‘one face with three layers’ is supposed that one
face has three layers, i.e. positive layer, negative layer and
middle layer. Positive layer is split by positive side’s solid.
Negative layer split by negative side’s solid. Middle layer
overlap positive and negative layer’s division. So sub-face of
middle layer connects sub-face of positive layer and negative
layer. A simple example is show in figure 4.
Figure 7. Middle layer of the ground surface is divided three
areas
4.3 Potential Application of ‘One Face with Three Layers’
Topology and data structure set by the data model of ‘one face
with three layers’ make solid adjacent inquire very easy, such as
above, below, left and right neighbor of a household in a
building. In mine hydrology-geology, inquire connection of a
containing water rock with mine laneway and exploitation area.
Every split line in positive layer and negative layer is connected
with a face. One face can be inquired connected with which face
by the splitting line. It needs to be noted that in 3D GIS, not all
face need to construction the topological relation of ‘one face
with three layers’. Only the faces such as ground face, fault or
floor face, which have the effect of compartmentation, need to
built ‘one face with three layer’. If abuse ‘one face with three
layer’, it may be not worth the candle.
Figure 4. One face with three layer
The positive is divided into two sub-faces of A1 and A2. The
negative layer is divided into two sub-faces of B1 and B2. The
middle layer is divided into four sub-faces of A2-B1, A2-B2,
A1-B1 and A1-B2.Now, we consider the ground-face (IJKLI) in
the model of figure 2. The positive layer is divided into two
sub-faces of land area (IMPLI) and water area (MJKPM) by
shore line (intersection line of water top face with the ground
face). It is show in figure 5.
4.4 Extension ‘One Face with Three Layers’ in 2D ----‘One
Line with Three Layers’
‘One face with 3 layers’ is a concept author proposed in study
3D topological relation by dint of 2D topological relation. Dose
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IAPRS, VOLUME XXXIV, PART 2, COMMISSION II, Xi’an, Aug. 20-23, 2002
exist similar concept in 2D GIS? The author think it dose exist
too. Now let us see figure 1 again. If AD is one freeway or
railway or large river, areas of 1,2,3,4 are district. We know that
freeway, railway and large river often make nature boundary of
district. In this case, concept of ‘one face with three layers’ can
be used. In 2D, we can call it as ‘one line with three layers’. So,
for linear object like freeway, railway and river, can construct
topological relation on ‘one line with three layer’, i.e. assume
the linear object has left layer, right layer and middle layer. Left
layer is split on left adjacent polygons, right layer is split on
right adjacent polygons. Middle layer overlap the left layer split
and right layer split, i.e. middle layer construct a relation of left
side’s split with right side’s split. In figure1, to construct
relation of ‘one line with three layers’ to line AD, the left layer
should be split to AC, CD two sections, right layer should be
split to AB, BD two sections, middle layer should be split to AB,
BC, CD three sections. It can be seen that middle layer’s split
scheme, is split scheme of traditional 2D GIS. The topological
relationship on the concept of ‘one line with three layer’ let the
integrality of a line object (like freeway, railway, river) can be
reserved, split of line left side and line right side are isolated,
through middle layer, left side and right side can be setup
linkage.
The structure of ‘one line with three layers’ is designed in
Appendex B.
ACKNOWLEDGEMENTS
The work is supported by Natural Science Foundation of China
(SES 49771059)and by Visitor Scholar Foundation of
Education Ministry Higher Education Key Laboratory.
REFERENCES
CHEN, X. and IKEDA K., 1994a. Raster dimensionl Modeling
of GIS Based on Delaunay Tetraheadal Tessellations, In:
International Archives of Photogrammetry and Remote Sensing,
Munich, Germany, 30,pp. 132-139.
CHEN, X. and IKEDA K., 1994b. Raster Algorithms for
Generating
Delaunay
Tetraheadal
Tessellations.
In:
International Archives of Photogrammetry and Remote Sensing,
Munich, Germany, 30, pp. 124-131.
Fritsch, D., 1996. Three-Dimentional Geographic Information
System: Status and Prospects. In: Karl Kraus, Peter Waldhausl,
ed. International Archives of Photogrammetry and Remote
Sensing, Vienna, Austria: 1996, XXXI(Part B3) .pp.215~221.
GONG, J. and XIA Zhongguo, 1997. An Integrated Data Model
in Three Dimensional GIS. Journal of Wuhan Technical
University of Surveying and Mapping, 22(1), pp. 7~15.
Traditional GIS data model is from part (arc) to entirety (linear
object), but ‘one line with three layers’ is from entirety (linear
object) to part (arc). In traditional model of point, line, polygon,
a linear object is divided a lot of arcs. If user wants to inquire a
river, it is needed to give the all arcs the same identify code. If
user wants to inquire a region in a river left side adjacent to
which region in river right side, the operation is very
complicated. If using structure of ‘one line with three layers’, it
can directly fetch left layer arc (virtual arc) left adjacent
polygon, and can easy through middle layer fetch river right
layer’s polygon, the inquire is simpler than traditional data
structure. The structure of ‘One line with three layers’ organises
isolated arcs into a linear object with topological relation which
considering the adjacent of left side and right side, getting a
more convenient inquire operation and fast inquire speed by a
very small index cost.
GUO, D., YANG Weiping, HAN Guojian, 1993. A Spatial and
Temporal 4-D Data Model for Mine GIS. ACTA GEODAETICA
et CARTOGRAPHICA SINICA, 1993, 22 (2), pp. 33~40
GUO, W. and CHEN Jun, 1997. The Formal Description of
Topological Spatial Relationship in 3D Based on Point Set
Topology. ACTA GEODAETICA et CARTOGRAPHICA
SINICA, 26(2), pp. 122~127.
GUO W. and CHEN Jun, 1998. A 3D Topological ER Model
Based on Space Partitioning. ACTA GEODAETICA et
CARTOGRAPHICA SINICA, 27(4), pp. 308~317.
HAN, G., GUO dazhi, JIN Xuelin, 1992. Octree Store and
Index Technique of Ore Body Information. ACTA
GEODAETICA et CARTOGRAPHICA SINICA, 21(1),
pp.13~17.
5. CONCLUSION
The authors from the view of boundary and coboundary in ISO
TC211 spatial schema, check five group relations formerly gave,
and
analyze
possible
problem
in
construct
‘Face-ExteriorRing-InteriorRings-PositiveSolid-NegativeSolid’,
then propose the concept of ‘One face with three layers’ (the
author called ‘One piece with three layers’ previously). The
corresponding data structure being designed. Then author
extend the concept of ‘one face with three layers’ to 2D,
propose the concept of ‘ one line with three layers’ to deal with
freeway, railway and river which as nature boundary of district,
hoping it can help linear object adjacent inquire.
Actually, ‘One face with three layer’ is added an index on
positive side sub-face and negative side sub-face, ‘One line with
three layer’ is added an index on left and right of a line. They all
are not obligatory relationship. If abusing it, it is possible to get
more kicks than halfpence. But for some dividing face or
dividing line, setup these adjunctive relationships, will
convenient to spatial topological inquire.
ISO TC211W2, 2001, ISO /DIS19017. Geographic Information,
Spatial schema.
LI, D. and LI Qingquan,1997. Study on a Hybrid Data Structure
in 3D GIS. ACTA GEODAETICA et CARTOGRAPHICA
SINICA, 26(2), pp. 128~133.
LI, R.X., 1994. Data Structure and Application Issues in 3-D
Geographic Information. GEOMATIC, 48(3), pp. 209~224.
LI, Q.Q. and LI Deren, 1998. Research on Conceptual Frame of
the Integration of 3D Spatial Data Model. ACTA GEODAETICA
et CARTOGRAPHICA SINICA , 27(4), pp. 325~329.
LI, Q.Y., 1996. 3D Vector Structure Topological Relationship
Research: [Doctor Paper]. Beijing: C.U.M.T.
LI, Q.Y., CAO Daiyong, GAO Wentai, 1996. 3D Vector
Structure Topological Relationship Based on Body Partition.
In: Xu Guanhua, Li Denren, Liu Xianlin, ed. Remote Sensing in
China.－－Thesis Collection of Commemorate China National
268
Qingyuan LI, Yanqing CHANG & Daiyong CAO
Remote Sensing Center 15 Anniversary. Beijing: The Surveying
and Mapping Press, pp.348~353.
The other classes defined as follows:
PosiChildFace: public ChildFace {// Child face of positive layer
SOLID *PosiSideSolid; // pointer of positive
// adjacent polyhedron
MidChildFace *MidChildFaces[ ]; // pointer array of middle
//layer child-faces
};
LI, Q.Y., CAO Daiyong ZHU Xiaodi, 1999. 3D Topology
Model on Body Divided. ISPRS, Vol. 32, Part 4 W12.
pp.91-98.
LI, Q.Y. and CAO Daiyong, 1997. Dynamic Setup and
Maintain of 3D GIS Topologic Relationship. ACTA
GEODAETICA et CARTOGRAPHICA SINICA. 26(3),
pp.235~240.
NegaChildFace: public ChildFace{//Child face of negative layer
SOLID *AdjacentSolid; // pointer of negative adjacent solid
MidChildFace *MidChildFaces[ ]; // pointer array of middle
// layer child-faces
};
LI, Q.Y. and CAO Daiyong, 1998a. 3D Topologic Relation of
3D GIS. In: Lai Pohchin, Leung Yee, Shi Wenzhong, ed.
Proceedings of International Conference on Modeling
Geigrophical and Environmental Systems with Geographical
Information Systems. Hong kong: Department of Grography
The Chinese University of Hong Kong. pp. 832~837.
We also can define a SideChildFace to replace PosiChildFace
and NegaChildFace.
SideChildFace : public ChildFace{//Child face of negative layer
SOLID
*AdaNegaSideSolid; // pointer of negative
// adjacent polyhedron
MidChildFace *MidChildFaces[ ]; // pointer array of middle
// layer child-faces
};
LI, Q.Y. and CAO Daiyong, 1998b. 3D Body Interpolation
and Application in 3D GIS. In: Lai Pohchin, Leung Yee, Shi
Wenzhong, ed. Proceedings of International Conference on
Modeling Geigrophical and Environmental Systems with
Geographical Information Systems. Hong kong: Department of
Grography The Chinese University of Hong Kong. pp.826~831.
Molenaar, M., 1990. A Formal Data Structure for Three
Dimensional Vector Maps. In Proceedings of 5th
Int.Symposium on Spatial Data Handling . Zurich, pp.830~843.
Morakot, P., Klaus Tempfli, Martien Molenaa, 1994. A
Tetrahedron-Based 3D Vector Data Model for Geoinformation.
In: Martien Molenaa, ed. Advanced geographic Data Modelling,
－ Spatial Data Modelling and Language for 2D and 3D
Application. Sylvia De Hoop, pp.129~140.
APPENDIX A. STRUCTURE OF ‘ONE FACE WITH
THREE LAYERS’
To realize concept of ‘One face with three layers’, the data
structure of C is designed as follows. Here only the data
members in the class are listed, all methods in the class are
omitted.
class FACE { // face class
ContourARC contourArcs[ ]; // array of contour lines
D3RING *ExteriorRing;
// pointer of Exterior ring
D3RING *InteriorRing[ ];
// array of interior rings
D2EDGE *PosiSideSplitEdges[ ];// pointer array of positive
// layer split edges
D2EDGE *NegaSideSplitEdges[]; // pointer array of negative
// layer split edges
PosiChildFace *PosiSideChildFaces[ ]; //pointer array of
// positive layer child-faces
NegaChildFace *NegaSideChildFaces[ ]; // pointer array of
// negative layer child-faces
//==== next 3 items can be omitted =======
MidChildFace
*midChildFaces[ ]; // pointer array of
//middle layer child-faces
SOLID *PositiveSolids[ ];
// pointer array of positive
// side adjacent polyhedrons
SOLID *NegativeSolids[ ] ; // pointer array of negative side
// adjacent polyhedrons
};
Child face of positive layer (PosiChildFace) and negative layer
(NegaChildFace) access other side solid by corresponding
middle layer child face. The class of middle layer child face is
defined as follow:
MidChildFace : public ChildFace{ //middle layer child face
SOLID *PosiSideSolid;//pointer of positive side adjacent solid
SOLID*NegaSideSolid;//pointer of negative side adjacent
solid
};
Positive layer, negative layer and middle layer access base face,
Exterior ring of the base face and begin position, end position of
contour arc in the base face by a common base class----abstract
child face class.
class ChildFace {
// abstract child Face
FACE
*BaseFace;
// pointer of base Face pointer
D3RING *ExteriorRing; // pointer of Exterior ring
ContourFromTo *ContourSegments[]; // pointer array
// of contour arc
};
ContourFromTo class is designed for searching start position
and end position in the arc of the base face. It can prevent store
contour arc in child face again, but can quickly get pattern
contour arc from its base face.
class ConrourFromTo { // contour arc
ContourARC *BaseContour; // the contour arc responding
// base contour arc in base face
D2VPOINT *From; // pointer of the contour begin point in
// base contour arc
D2VPOINT *To; // pointer of the contour end point in
// base contour arc
};
APPENDIX B. STRUCTURE OF ‘ONE LINE WITH
THREE LAYERS’
class LINE{
NODE
*Begin;
NODE
*End;
VirtualARC *LeftARCs[ ];
VirtualARC *RightARCs[ ];
In FACE class, we can also not store midChildFaces,
PositiveSolids and NegativeSolids, but access them by
PosiSideChildFaces and NegaSideChildFaces.
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//linear object
// begin node pointer
// end node pointer
// array of left arc pointer
// array of right arc pointer
IAPRS, VOLUME XXXIV, PART 2, COMMISSION II, Xi’an, Aug. 20-23, 2002
ARC
MiddleARCs[ ];
// array of middle arc
};
class VirtualARC {
// virtual arc
NODE
*Begin;
// begin node pointer
NODE
*End;
// end node pointer
ARC
*Arcs[ ];
// array of middle arc pointer
POLYGON *AdjacentPolygon; // adjacent polygon pointer,
// left arc is left adjacent polygon,
// right arc is right adjacent polygon
};
class ARC {
// real arc
NODE
*Begin;
// begin node pointer
NODE
*End;
// end node pointer
POLYGON *LeftPolygon; // left adjacent polygon
POLYGON *RightPolygon; // right adjacent polygon
POINT
Points[ ];
// pattern point of arc
};
270